Strong spatial mixing in homomorphism spaces

Given a countable graph G and a finite graph H, we consider Hom(G, H) the set of graph homomorphisms from G to H and we study Gibbs measures supported on Hom(G, H). We develop some sufficient and other necessary conditions for the existence of Gibbs specifications on Hom(G, H) satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph H has a combinatorial property called dismantlability iff for every G of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that such Gibbs specification can be chosen to have weak spatial mixing. In addition, we exhibit a subfamily of graphs H for which there exists Gibbs specifications satisfying strong spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs specification has strong spatial mixing.